Question

Under some circumstances, a star can collapse into an extremely dense object made mostly of neutrons and called a neutron star. The density of a neutron star is roughly 10^14 times as great as that of ordinary solid matter. Suppose we represent the star as a uniform, solid, rigid sphere, both before and after the collapse. The star's initial radius was 7.0×10^5 km (comparable to our sun); its final radius is 18km .

If the original star rotated once in 35 days, find the angular speed of the neutron star.

Answer 1

Concepts and reason

The concepts used to solve this problem are time period, angular velocity, moment of inertia and angular momentum

First calculate the time period of the original star in seconds, and then use this time period to calculate the initial angular speed of original star.

Use mass and perpendicular distance relation to calculate the initial and final moment of inertia the star.

Use the relation of moment of inertia and angular velocity to calculate the angular momentum.

Finally use conservation of momentum to calculate the angular velocity of the neutron star.

Fundamentals

Expression for the angular speed of the star is,

$=0 $

Here, is the angular velocity and is the time period.

Expression for the moment of inertia of the star is,

Here, is the moment of inertia, is the mass of the star, and is the radius of the star.

Expression for the angular momentum for the star is,

$L=10 $

Here, is the angular momentum.

The original star rotated once in 35 days.

Expression for the angular velocity of the original star is,

Here, is the angular velocity of the original star.

Substitute 35 days for .

Two stars are of equal masses.

Expression for the moment of inertia of original star is,

Here, is the initial moment of inertia of original star and is the radius of the original star.

Expression for the moment of inertia of neutron star is,

Here, is the moment of inertia of neutron star and is the radius of the neutron star.

From conservation of momentum,

Here, is the angular velocity of the neutron star.

Substitute for and for .

Rearrange the equation to get angular velocity of the neutron star.

Expression for the angular velocity of the neutron star is,

Substitute for , for , and for .

Ans:

The angular velocity of the neutron star is .